Find the Gini index for the given Lorentz curve. Round your answer to four decimal places, if necessary. $L ( x ) = 0.7 x ^ { 2 } + 0.3 x$
A) 1.7667
B) 0.2333
C) 0.2000
D) 0.2800

Question

Quizplus · Accepted Answer

The Gini index can be calculated using the formula $G = 1 - 2 \int_{0}^{1} L(x) dx$. Substituting $L(x) = 0.7x^2 + 0.3x$ into the formula, we get $G = 1 - 2 \int_{0}^{1} (0.7x^2 + 0.3x) dx$. Integrating, we find $G = 1 - 2 \left[\frac{0.7x^3}{3} + \frac{0.3x^2}{2}\right]_0^1 = 1 - 2 \left[\frac{0.7}{3} + \frac{0.3}{2}\right] = 1 - 2 \left[\frac{7}{30} + \frac{3}{20}\right] = 1 - 2 \left[\frac{14 + 9}{60}\right] = 1 - 2 \left[\frac{23}{60}\right] = 1 - \frac{23}{30} = \frac{7}{30} = 0.2333$.

Find the Gini Index for the Given Lorentz Curve L(x)=0.7x2+0.3xL ( x ) = 0.7 x ^ { 2 } + 0.3 xL(x)=0.7x2+0.3x

Find the Gini Index for the Given Lorentz Curve $L ( x ) = 0.7 x ^ { 2 } + 0.3 x$